The optimal bandwidth \(h^\ast _N\)  in Equation (REF ) is nice for theoretical purposes but is not applicable in practice due to the simple fact that it depends on the unknown spectral density \(f\) .
Instead, many methods to approximate the optimal bandwidth can be found in literature. Scott's or Silverman's rules of thumb are widespread in practices for their ease of use but assume the sample to be drawn from a Gaussian distribution [1]}.
These methods are efficient and fast but their accuracy can only be guaranteed in the Gaussian case or for the estimation of unimodal and close to Gaussian densities.
When the form of the sought-after density is unknown, which is usually the case, methods like the unbiased cross-validation [2]}, [3]} or the Sheather and Jones plug-in method [4]} are far better suited.
There are many more cross-validation and plug-in methods at hand, see e.g. [5]} for an overview, but we applied the Sheather and Jones method as it is already implemented in R. Similar results were achieved with other methods.
